The [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson(<math> \lambda </math>) distribution]] can be approximated with [[AP_Statistics_Curriculum_2007_Normal_Prob |Normal]] when <math> \lambda </math> is large.

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The [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson(<math> \lambda </math>) Distribution]] can be approximated with [[AP_Statistics_Curriculum_2007_Normal_Prob |Normal]] when <math> \lambda </math> is large.

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For sufficiently large values of λ, (say λ>1,000), the [[AP_Statistics_Curriculum_2007_Normal_Prob |Normal(<math>\mu=\lambda, \sigma^2=\lambda</math>)]] distribution is an excellent approximation to the [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson(λ)]] distribution. If λ is greater than about 10, then the normal distribution is a good approximation if an appropriate [http://en.wikipedia.org/wiki/Continuity_correction continuity correction] is performed.

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For sufficiently large values of λ, (say λ>1,000), the [[AP_Statistics_Curriculum_2007_Normal_Prob |Normal(<math>\mu=\lambda, \sigma^2=\lambda</math>)]] Distribution is an excellent approximation to the [[AP_Statistics_Curriculum_2007_Distrib_Poisson | Poisson(λ)]] Distribution. If λ is greater than about 10, then the Normal Distribution is a good approximation if an appropriate [http://en.wikipedia.org/wiki/Continuity_correction continuity correction] is performed.

If <math>x_o</math> is a non-negative integer, <math>X\sim Poisson(\lambda)</math> and <math>U\sim Normal(\mu=\lambda, \sigma^2=\lambda</math>), then <math>P_X(X<x_o) = P_U(U<x_o+0.5)</math>.

If <math>x_o</math> is a non-negative integer, <math>X\sim Poisson(\lambda)</math> and <math>U\sim Normal(\mu=\lambda, \sigma^2=\lambda</math>), then <math>P_X(X<x_o) = P_U(U<x_o+0.5)</math>.

Normal Approximation to Poisson Distribution

For sufficiently large values of λ, (say λ>1,000), the Normal(μ = λ,σ2 = λ) Distribution is an excellent approximation to the Poisson(λ) Distribution. If λ is greater than about 10, then the Normal Distribution is a good approximation if an appropriate continuity correction is performed.

Examples

Suppose cars arrive at a parking lot at a rate of 50 per hour. Let’s assume that the process is a Poisson random variable with λ = 50. Compute the probability that in the next hour the number of cars that arrive at this parking lot will be between 54 and 62. We can compute this as follows:
The figure below from SOCR shows this probability.

Note: We observe that this distribution is bell-shaped. We can use the normal distribution to approximate this probability. Using , together with the continuity correction for better approximation we obtain , which is close to the exact that was found earlier. The figure below shows this probability.